Problem 34 - Algebra and Trigonometry

Question

In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ \tan (-t)-\tan t $$

Step-by-Step Solution

Answer

The expression \( \tan(-t) - \tan(t) \) simplifies to \( -2c \) in terms of \( a, b, c \).

Step 1: Recognize the trigonometric identity

Knowing that \( \tan(-t) = -\tan(t) \) helps us in simplifying the given equation. By substituting this identity into our expression, we obtain \( -\tan(t) - \tan(t) \).

Step 2: Simplify the expression

The equation simplifies to \( -2 \tan(t) \).

Step 3: Substitute variables

In the end, replace \( \tan(t) \) with \( c \), so we get the final expression to be \( -2c \).

Key Concepts

Sine

The sine function, commonly denoted as \( \sin \), is an essential trigonometric function that represents the y-coordinate of a point on the unit circle. The unit circle is a circle with radius 1, centered at the origin of the coordinate system. As the angle \( t \) changes, the point moves along the circle, and the sine value changes accordingly. This makes the sine function important for describing periodic phenomena, such as sound waves and light waves.

Key characteristics of the sine function include:

• Range: The sine of an angle is always between -1 and 1.
• Periodicity: The sine function repeats every \( 2\pi \) radians due to its circular nature. This means \( \sin(t + 2\pi) = \sin(t) \).
• Odd Function: Sine is an odd function, meaning that \( \sin(-t) = -\sin(t) \). This property is useful for simplifying expressions, similar to the identity used in solving the exercise.

The sine function's ability to model oscillating patterns makes it invaluable in physics, engineering, and various other fields.

Cosine

Cosine, represented as \( \cos \), is another fundamental trigonometric function that works in tandem with sine. It describes the x-coordinate of a point on the unit circle as the angle \( t \) changes. Along with sine, the cosine function is critical in analyzing waveforms and circular motion across diverse scientific disciplines.

Understanding the properties of cosine:

• Range: Like sine, cosine values also lie between -1 and 1.
• Periodicity: Cosine shares the same basic periodicity as sine, repeating every \( 2\pi \) radians: \( \cos(t + 2\pi) = \cos(t) \).
• Even Function: Cosine is an even function, which means \( \cos(-t) = \cos(t) \). This symmetry about the y-axis can simplify calculations and transformations in trigonometric expressions.

Cosine's properties allow it to handle phase shifts and amplitude changes smoothly, making it integral to fields like electrical engineering and signal processing.

Tangent

The tangent function, indicated as \( \tan \), is the ratio of sine to cosine: \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). It is a pivotal function in trigonometry and is often used to model slopes and angles of elevation.

Some important features of the tangent function include:

• Undefined Points: Tangent is undefined whenever \( \cos(t) = 0 \), meaning at odd multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \)
• Periodicity: Unlike sine and cosine, tangent repeats every \( \pi \) radians: \( \tan(t + \pi) = \tan(t) \).
• Odd Function: Tangent is an odd function, which means \( \tan(-t) = -\tan(t) \). This property was specifically utilized in the original exercise to simplify the expression \( \tan(-t) - \tan(t) \).

Due to its unique properties, the tangent function is essential in problems involving real-world measurement and optimization, from calculating angles in architecture to adjusting components in control systems.